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Data re-uploading and expressivity/trainability trade-off on quantum policy gradients

Bernardo Manuel Barbosa Soares

Published Dec 1, 2024• 0 citations
OpenAlex

Abstract

This thesis investigates the integration of quantum computing into machine learning, specifically focusing on reinforcement learning. The study focuses on the building of a code library to enable the exploration of the use of variational quantum circuits as function approximators within reinforcement learning algorithms, specifically policy gradient methods. A comprehensive framework is developed to analyze the performance and trainability of variational quantum circuit-based reinforcement learning models, supported by this library, to understand how different quantum circuit designs and data re-uploading techniques might affect the performance and trainability. To demonstrate the library’s capabilities, experiments were conducted in classic control environments like CartPole-v1 and Acrobot-v1, evaluating the performance (measured as average return) and trainability of three quantum models. These experiments examined the effects of data re-uploading, classical postprocessing, and inverse temperature parameter scheduling on both metrics. The empirical results show that variational quantum circuits are effective function approximators, with data re-uploading significantly enhancing model performance. Additionally, the globality of the measured observables plays a critical role, with higher globality values leading to better performance. Despite increasing circuit expressiveness, the gradient magnitudes and variances remain substantial throughout training. Furthermore, the tested inverse temperature schedules, used to adjust the policy’s level of exploration, did not show any significant advantage, indicating the challenge in finding optimal settings for these schedules. The study also introduces the universal quantum classifier adapted for reinforcement learning. Contrary to the barren plateau phenomenon, it has been empirically demonstrated that although increasing the number of qubits leads to performance drops in the Acrobot environment, it does not lead to vanishing gradients during the agents’ training process. Furthermore, under specific sets of hyperparameters, all three quantum models demonstrate strong and similar performance levels in both environments.

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This thesis investigates the integration of quantum computing into machine learning, specifically focusing on reinforcement learning. The study focuses on the building of a code library to enable the exploration of the use of variational quantum circuits as function approximators within reinforcement learning algorithms, specifically policy gradient methods. A comprehensive framework is developed to analyze the performance and trainability of variational quantum circuit-based reinforcement learning models, supported by this library, to understand how different quantum circuit designs and data re-uploading techniques might affect the performance and trainability. To demonstrate the library’s capabilities, experiments were conducted in classic control environments like CartPole-v1 and Acrobot-v1, evaluating the performance (measured as average return) and trainability of three quantum models. These experiments examined the effects of data re-uploading, classical postprocessing, and inverse temperature parameter scheduling on both metrics. The empirical results show that variational quantum circuits are effective function approximators, with data re-uploading significantly enhancing model performance. Additionally, the globality of the measured observables plays a critical role, with higher globality values leading to better performance. Despite increasing circuit expressiveness, the gradient magnitudes and variances remain substantial throughout training. Furthermore, the tested inverse temperature schedules, used to adjust the policy’s level of exploration, did not show any significant advantage, indicating the challenge in finding optimal settings for these schedules. The study also introduces the universal quantum classifier adapted for reinforcement learning. Contrary to the barren plateau phenomenon, it has been empirically demonstrated that although increasing the number of qubits leads to performance drops in the Acrobot environment, it does not lead to vanishing gradients during the agents’ training process. Furthermore, under specific sets of hyperparameters, all three quantum models demonstrate strong and similar performance levels in both environments.
Quantum Computing aims to streamline machine learning, making it more effective with fewer trainable parameters. This reduction of parameters can speed up the learning process and reduce the use of computational resources. However, in the current phase of quantum computing development, known as the noisy intermediate-scale quantum era (NISQ), learning is difficult due to a limited number of qubits and widespread quantum noise. To overcome these challenges, researchers are focusing on variational quantum circuits (VQCs). VQCs are hybrid algorithms that merge a quantum circuit, which can be adjusted through parameters, with traditional classical optimization techniques. These circuits require only few qubits for effective learning. Recent studies have presented new ways of applying VQCs to reinforcement learning, showing promising results that warrant further exploration. This study investigates the effects of various techniques — data re-uploading, input scaling, output scaling — and introduces exponential learning rate decay in the quantum proximal policy optimization algorithm’s actor-VQC. We assess these methods in the popular Frozen Lake and Cart Pole environments. Our focus is on their ability to reduce the number of parameters in the VQC without losing effectiveness. Our findings indicate that data re-uploading and an exponential learning rate decay significantly enhance hyperparameter stability and overall performance. While input scaling does not improve parameter efficiency, output scaling effectively manages greediness, leading to increased learning speed and robustness.
Variational quantum circuits have recently gained popularity as quantum machine learning models. While considerable effort has been invested to train them in supervised and unsupervised learning settings, relatively little attention has been given to their potential use in reinforcement learning. In this work, we leverage the understanding of quantum policy gradient algorithms in a number of ways. First, we investigate how to construct and train reinforcement learning policies based on variational quantum circuits. We propose several designs for quantum policies, provide their learning algorithms, and test their performance on classical benchmarking environments. Second, we show the existence of task environments with a provable separation in performance between quantum learning agents and any polynomial-time classical learner, conditioned on the widely-believed classical hardness of the discrete logarithm problem. We also consider more natural settings, in which we show an empirical quantum advantage of our quantum policies over standard neural-network policies. Our results constitute a first step towards establishing a practical near-term quantum advantage in a reinforcement learning setting. Additionally, we believe that some of our design choices for variational quantum policies may also be beneficial to other models based on variational quantum circuits, such as quantum classifiers and quantum regression models.
Variational Quantum Circuits are being used as versatile Quantum Machine Learning models. Some empirical results exhibit an advantage in supervised and generative learning tasks. However, when applied to Reinforcement Learning, less is known. In this work, we considered a Variational Quantum Circuit composed of a low-depth hardware-efficient ansatz as the parameterized policy of a Reinforcement Learning agent. We show that an εεε-approximation of the policy gradient can be obtained using a logarithmic number of samples concerning the total number of parameters. We empirically verify that such quantum models behave similarly or even outperform typical classical neural networks used in standard benchmarking environments and in quantum control, using only a fraction of the parameters. Moreover, we study the Barren Plateau phenomenon in quantum policy gradients using the Fisher Information Matrix spectrum.
Abstract Variational quantum circuits are being used as versatile quantum machine learning models. Some empirical results exhibit an advantage in supervised and generative learning tasks. However, when applied to reinforcement learning, less is known. In this work, we considered a variational quantum circuit composed of a low-depth hardware-efficient ansatz as the parameterized policy of a reinforcement learning agent. We show that an 𝜖-approximation of the policy gradient can be obtained using a logarithmic number of samples concerning the total number of parameters. We empirically verify that such quantum models behave similarly to typical classical neural networks used in standard benchmarking environments and quantum control, using only a fraction of the parameters. Moreover, we study the barren plateau phenomenon in quantum policy gradients using the Fisher information matrix spectrum.
Quantum re-uploading models have been extensively investigated as a form of machine learning within the context of variational quantum algorithms. Their trainability and expressivity are not yet fully understood and are critical to their performance. In this work, we address trainability through the lens of the magnitude of the gradients of the cost function. We prove bounds for the differences between gradients of the better-studied data-less parameterized quantum circuits and re-uploading models. We coin the concept of absorption witness to quantify such difference. For the expressivity, we prove that quantum re-uploading models output functions with vanishing high-frequency components and upper-bounded derivatives with respect to data. As a consequence, such functions present limited sensitivity to fine details, which protects against overfitting. We performed numerical experiments extending the theoretical results to more relaxed and realistic conditions. Overall, future designs of quantum re-uploading models will benefit from the strengthened knowledge delivered by the uncovering of absorption witnesses and vanishing high frequencies.
The comparative evaluation between classical and quantum reinforcement learning (QRL) paradigms was conducted to investigate their convergence behavior, robustness under observational noise, and computational efficiency in a benchmark control environment. The study employed a multilayer perceptron (MLP) agent as a classical baseline and a parameterized variational quantum circuit (VQC) as a quantum counterpart, both trained on the CartPole-v1 environment over 500 episodes. Empirical results demonstrated that the classical MLP achieved near-optimal policy convergence with a mean return of 498.7 +/- 3.2, maintaining stable equilibrium throughout training. In contrast, the VQC exhibited limited learning capability, with an average return of 14.6 +/- 4.8, primarily constrained by circuit depth and qubit connectivity. Noise robustness analysis further revealed that the MLP policy deteriorated gracefully under Gaussian perturbations, while the VQC displayed higher sensitivity at equivalent noise levels. Despite the lower asymptotic performance, the VQC exhibited significantly lower parameter count and marginally increased training time, highlighting its potential scalability for low-resource quantum processors. The results suggest that while classical neural policies remain dominant in current control benchmarks, quantum-enhanced architectures could offer promising efficiency advantages once hardware noise and expressivity limitations are mitigated.